As of Decibel 1.2 it is possible to use a receipt obtained by purchasing Decibel through the Mac App Store (MAS) to unlock the direct-download version. The process will be automatic once Apple approves a newer version of Decibel on the MAS, but until then it can be accomplished manually with the following steps:

Locate the copy of Decibel from the MAS, and copy the Contents/_MASReceipt/receipt file into the folder created in step 1. You'll need to use the "Show Package Contents" menu item to look inside the app itself.

Expanded directions:

Open a new Finder window

Press command-shift-H to go to your home folder

Open the Library Folder

Open the Application Support folder

Open the Decibel folder

Press command-shift-N to create a new folder, and call it App Store Receipts

Open the App Store Receipts folder

Leave this window open but move it aside (you'll copy the receipt into this folder later)